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Refinements, extensions and generalizations of the second Kershaw’s double inequality. (English) Zbl 1155.26009

Let \(\Psi(x)=\Gamma'(x) / \Gamma(x)\), where \(\Gamma\) is the classical gamma function. The second Kershaw’s double inequality reads as follows: \[ \exp[(1-s)\Psi(x+\sqrt{s})]<\frac{\Gamma(x+1)}{\Gamma(x+s)}<\exp[(1-s) \Psi(x+\frac{s+1}{2})], \] where \(0<s<1\) and \(x\geq 1\). It was subject of many investigations; the bibliography of the present paper contains 54 titles. The authors generalize, extend and refine the right-hand side inequality, as well as other related inequalities.

MSC:

26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations
26D20 Other analytical inequalities
33B10 Exponential and trigonometric functions
33B15 Gamma, beta and polygamma functions
65R10 Numerical methods for integral transforms
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